With a CAt22 I don't know..I have a friend that crossed the Atlantic in a 24 foot boat (and he will not do it again), and another one that sailed from France to Casablanca and to the Canaries on an aluminium Pen Duick 20 footer..
With the righet weather wind, you can go as far as you want.
I'm docked in Milford, CT. Sail the LI Sound all the time. Can get a little nasty in a 22 but generally manageable and you're never far from a cozy marina. South shore...don't know but would think if you avoid the open seas you'll be fine. Take your time and have fun.
The South Shore bays have some low fixed bridges (15' or so behind Long Beach among others) that might discourage you from going that route. Some bridges open and some don't - check the chart.
While the southern bays generally have better wind off the ocean then you get on the north shore in the summer there are quite a few spots with skinny water.
Entering or leaving the south shore inlets can also be dicey if the tide is ebbing - they can generate some intense waves when the tide flows into the ocean swells even on a fair weather day. Near slack tide or the beginning of flood are probably better times to use these inlets. Tidal races form around all the south shore inlets creating some strong currents.
I am not sure which direction 'around' you are contemplating but because of our usual SW summer winds most would choose to go down the East River first and then run/beam reach out to Montauk. Then you can tack back up the Sound at your leisure. Even the ALIR (Yacht Scoring - A complete web based regatta administration and yacht scoring program ) goes around the island in this direction. I wonder if you have considered trying to sign on as a crew member for this?